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Irrational Numbers and Rounding Off*

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Irrational Numbers and Rounding Off* OpenStax-CNX module: m31341 1 Irrational Numbers and Rounding Off* Rory Adams Free High School Science Texts Project Mark Horner Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 1 Introduction You have seen that repeating decimals may take a lot of paper and ink to write out. Not only is that impossible, but writing numbers out to many decimal places or a high accuracy is very inconvenient and rarely gives practical answers. For this reason we often estimate the number to a certain number of decimal places or to a given number of signicant gures, which is even better. 2 Irrational Numbers Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. This means that any number that is not a terminating decimal number or a repeating decimal number is irrational. Examples of irrational numbers are: p p p 2; 3; 3 4; π; p (1) 1+ 5 2 ≈ 1; 618 033 989 tip: When irrational numbers are written in decimal form, they go on forever and there is no repeated pattern of digits. If you are asked to identify whether a number is rational or irrational, rst write the number in decimal form. If the number is terminated then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational. When you write irrational numbers in decimal form, you may (if you have a lot of time and paper!) continue writing them for many, many decimal places. However, this is not convenient and it is often necessary to round o. *Version 1.4: Apr 20, 2011 7:31 am -0500 http://creativecommons.org/licenses/by/3.0/ http://cnx.org/content/m31341/1.4/ OpenStax-CNX module: m31341 2 2.1 Investigation : Irrational Numbers Which of the following cannot be written as a rational number? Remember: A rational number is a fraction with numerator and denominator as integers. Terminating decimal numbers or repeating decimal numbers are rational. 1. π = 3; 14159265358979323846264338327950288419716939937510::: 2. 1,4 3. 1; 618 033 989 ::: 4. 100 3 Rounding O Rounding o or approximating a decimal number to a given number of decimal places is the quickest way to approximate a number. For example, if you wanted to round-o 2; 6525272 to three decimal places then you would rst count three places after the decimal. 2; 652j5272 (2) All numbers to the right of j are ignored after you determine whether the number in the third decimal place must be rounded up or rounded down. You round up the nal digit if the rst digit after the j was greater or equal to 5 and round down (leave the digit alone) otherwise. In the case that the rst digit before the j is 9 and the j you need to round up the 9 becomes a 0 and the second digit before the j is rounded up. So, since the rst digit after the j is a 5, we must round up the digit in the third decimal place to a 3 and the nal answer of 2; 6525272 rounded to three decimal places is 2; 653 (3) Exercise 1: Rounding-O (Solution on p. 4.) Round-o the following numbers to the indicated number of decimal places: 1. 120 _ _ to 3 decimal places 99 = 1; 21212121212 2. to 4 decimal places pπ = 3; 141592654::: 3. 3 = 1; 7320508::: to 4 decimal places 4 Summary • Irrational numbers are numbers that cannot be written as a fraction with the numerator and denomi- nator as integers. • For convenience irrational numbers are often rounded o to a specied number of decimal places 5 End of Chapter Exercises 1. Write the following rational numbers to 2 decimal places: a. 1 2 b. 1 c. 0; 111111 d. 0; 999991 http://cnx.org/content/m31341/1.4/ OpenStax-CNX module: m31341 3 Click here for the solution1 2. Write the following irrational numbers to 2 decimal places: a. 3; 141592654::: b. 1; 618 033 989 ::: c. 1; 41421356::: d. 2; 71828182845904523536::: Click here for the solution2 3. Use your calculator and write the following irrational numbers to 3 decimal places: p a. p2 b. p3 c. p5 d. 6 Click here for the solution3 4. Use your calculator (where necessary) and write the following irrational numbers to 5 decimal places: p a. p8 b. p768 c. p100 d. p0; 49 e. p0; 0016 f. p0; 25 g. p36 h. p1960 i. 0p; 0036 j. −p8 0; 04 k. 5 80 Click here for the solution4 5. Write the following irrational numbers to 3 decimal places and then write them as a rational number p to get an approximation to the irrational number. For example, . To 3 decimal places, p p 3 = 1; 73205::: . 732 183 . Therefore, is approximately 183 . 3 = 1; 732 1; 732 = 1 1000 = 1 250 3 1 250 a. 3; 141592654::: b. 1; 618 033 989 ::: c. 1; 41421356::: d. 2; 71828182845904523536::: Click here for the solution5 1See the le at <http://cnx.org/content/m31341/latest/http://www.fhsst.org/llN> 2See the le at <http://cnx.org/content/m31341/latest/http://www.fhsst.org/llR> 3See the le at <http://cnx.org/content/m31341/latest/http://www.fhsst.org/lln> 4See the le at <http://cnx.org/content/m31341/latest/http://www.fhsst.org/llQ> 5See the le at <http://cnx.org/content/m31341/latest/http://www.fhsst.org/llU> http://cnx.org/content/m31341/1.4/ OpenStax-CNX module: m31341 4 Solutions to Exercises in this Module Solution to Exercise (p. 2) Step 1. a. 120 _ _ 99 = 1; 212j12121212 b. pπ = 3; 1415j92654::: c. 3 = 1; 7320j508::: Step 2. a. The last digit of 120 _ _ must be rounded-down. 99 = 1; 212j12121212 b. The last digit of must be rounded-up. pπ = 3; 1415j92654::: c. The last digit of 3 = 1; 7320j508::: must be rounded-up. Step 3. a. 120 rounded to 3 decimal places 99 = 1; 212 b. rounded to 4 decimal places pπ = 3; 1416 c. 3 = 1; 7321 rounded to 4 decimal places http://cnx.org/content/m31341/1.4/.
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